Graeme Henkelman

A climbing image nudged elastic band method for finding saddle points and minimum energy paths

A modification of the nudged elastic band method for finding minimum energy paths is presented. One of the images is made to climb up along the elastic band to converge rigorously on the highest saddle point. Also, variable spring constants are used to increase the density of images near the top of the energy barrier to get an improved estimate of the reaction coordinate near the saddle point. Applications to CH4 dissociative adsorption on Ir~111! and H2 on Si~100! using plane wave based density functional theory are presented.

Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points

An improved way of estimating the local tangent in the nudged elastic band method for finding minimum energy paths is presented. In systems where the force along the minimum energy path is large compared to the restoring force perpendicular to the path and when many images of the system are included in the elastic band, kinks can develop and prevent the band from converging to the minimum energy path. We show how the kinks arise and present an improved way of estimating the local tangent which solves the problem. The task of finding an accurate energy and configuration for the saddle point is also discussed and examples given where a complementary method, the dimer method, is used to efficiently converge to the saddle point. Both methods only require the first derivative of the energy and can, therefore, easily be applied in plane wave based density-functional theory calculations. Examples are given from studies of the exchange diffusion mechanism in a Si crystal, Al addimer formation on the Al(100) surfa...

A grid-based Bader analysis algorithm without lattice bias

A computational method for partitioning a charge density grid into Bader volumes is presented which is efficient, robust, and scales linearly with the number of grid points. The partitioning algorithm follows the steepest ascent paths along the charge density gradient from grid point to grid point until a charge density maximum is reached. In this paper, we describe how accurate off-lattice ascent paths can be represented with respect to the grid points. This improvement maintains the efficient linear scaling of an earlier version of the algorithm, and eliminates a tendency for the Bader surfaces to be aligned along the grid directions. As the algorithm assigns grid points to charge density maxima, subsequent paths are terminated when they reach previously assigned grid points. It is this grid-based approach which gives the algorithm its efficiency, and allows for the analysis of the large grids generated from plane-wave-based density functional theory calculations.

Improved grid‐based algorithm for Bader charge allocation

An improvement to the grid‐based algorithm of Henkelman et al. for the calculation of Bader volumes is suggested, which more accurately calculates atomic properties as predicted by the theory of Atoms in Molecules. The CPU time required by the improved algorithm to perform the Bader analysis scales linearly with the number of interatomic surfaces in the system. The new algorithm corrects systematic deviations from the true Bader surface, calculated by the original method and also does not require explicit representation of the interatomic surfaces, resulting in a more robust method of partitioning charge density among atoms in the system. Applications of the method to some small systems are given and it is further demonstrated how the method can be used to define an energy per atom in ab initio calculations.

A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives

The problem of determining which activated (and slow) transitions can occur from a given initial state at a finite temperature is addressed. In the harmonic approximation to transition state theory this problem reduces to finding the set of low lying saddle points at the boundary of the potential energy basin associated with the initial state, as well as the relevant vibrational frequencies. Also, when full transition state theory calculations are carried out, it can be useful to know the location of the saddle points on the potential energy surface. A method for finding saddle points without knowledge of the final state of the transition is described. The method only makes use of first derivatives of the potential energy and is, therefore, applicable in situations where second derivatives are too costly or too tedious to evaluate, for example, in plane wave based density functional theory calculations. It is also designed to scale efficiently with the dimensionality of the system and can be applied to ve...

Optimization methods for finding minimum energy paths.

A comparison of chain-of-states based methods for finding minimum energy pathways (MEPs) is presented. In each method, a set of images along an initial pathway between two local minima is relaxed to find a MEP. We compare the nudged elastic band (NEB), doubly nudged elastic band, string, and simplified string methods, each with a set of commonly used optimizers. Our results show that the NEB and string methods are essentially equivalent and the most efficient methods for finding MEPs when coupled with a suitable optimizer. The most efficient optimizer was found to be a form of the limited-memory Broyden-Fletcher-Goldfarb-Shanno method in which the approximate inverse Hessian is constructed globally for all images along the path. The use of a climbing-image allows for finding the saddle point while representing the MEP with as few images as possible. If a highly accurate MEP is desired, it is found to be more efficient to descend from the saddle to the minima than to use a chain-of-states method with many images. Our results are based on a pairwise Morse potential to model rearrangements of a heptamer island on Pt(111), and plane-wave based density functional theory to model a rollover diffusion mechanism of a Pd tetramer on MgO(100) and dissociative adsorption and diffusion of oxygen on Au(111).

Comparison of methods for finding saddle points without knowledge of the final states

Within the harmonic approximation to transition state theory, the biggest challenge involved in finding the mechanism or rate of transitions is the location of the relevant saddle points on the multidimensional potential energy surface. The saddle point search is particularly challenging when the final state of the transition is not specified. In this article we report on a comparison of several methods for locating saddle points under these conditions and compare, in particular, the well-established rational function optimization (RFO) methods using either exact or approximate Hessians with the more recently proposed minimum mode following methods where only the minimum eigenvalue mode is found, either by the dimer or the Lanczos method. A test problem involving transitions in a seven-atom Pt island on a Pt(111) surface using a simple Morse pairwise potential function is used and the number of degrees of freedom varied by varying the number of movable atoms. In the full system, 175 atoms can move so 525 degrees of freedom need to be optimized to find the saddle points. For testing purposes, we have also restricted the number of movable atoms to 7 and 1. Our results indicate that if attempting to make a map of all relevant saddle points for a large system (as would be necessary when simulating the long time scale evolution of a thermal system) the minimum mode following methods are preferred. The minimum mode following methods are also more efficient when searching for the lowest saddle points in a large system, and if the force can be obtained cheaply. However, if only the lowest saddle points are sought and the calculation of the force is expensive but a good approximation for the Hessian at the starting position of the search can be obtained at low cost, then the RFO approaches employing an approximate Hessian represent the preferred choice. For small and medium sized systems where the force is expensive to calculate, the RFO approaches employing an approximate Hessian is also the more efficient, but when the force and Hessian can be obtained cheaply and only the lowest saddle points are sought the RFO approach using an exact Hessian is the better choice. These conclusions have been reached based on a comparison of the total computational effort needed to find the saddle points and the number of saddle points found for each of the methods. The RFO methods do not perform very well with respect to the latter aspect, but starting the searches further away from the initial minimum or using the hybrid RFO version presented here improves this behavior considerably in most cases.

CO Oxidation Mechanism on CeO2-Supported Au Nanoparticles

Density functional theory was used to study the CO oxidation catalytic activity of CeO(2)-supported Au nanoparticles (NPs). Experimental observations on CeO(2) show that the surface of CeO(2) is enriched with oxygen vacancies. We compare CO oxidation by a Au(13) NP supported on stoichiometric CeO(2) (Au(13)@CeO(2)-STO) and partially reduced CeO(2) with three vacancies (Au(13)@CeO(2)-3VAC). The structure of the Au(13) NP was chosen to minimize structural rearrangement during CO oxidation. We suggest three CO oxidation mechanisms by Au(13)@CeO(2): CO oxidation by coadsorbed O(2), CO oxidation by a lattice oxygen in CeO(2), and CO oxidation by O(2) bound to a Au-Ce(3+) anchoring site. Oxygen vacancies are shown to open a new CO oxidation pathway by O(2) bound to a Au-Ce(3+) anchoring site. Our results provide a design strategy for CO oxidation on supported Au catalysts. We suggest lowering the vacancy formation energy of the supporting oxide, and using an easily reducible oxide to increase the concentration of reduced metal ions, which act as anchoring sites for O(2) molecules.

Long time scale kinetic Monte Carlo simulations without lattice approximation and predefined event table

We present a method for carrying out long time scale dynamics simulations within the harmonic transition state theory approximation. For each state of the system, characterized by a local minimum on the potential energy surface, multiple searches for saddle points are carried out using random initial directions. The dimer method is used for the saddle point searches and the rate for each transition mechanism is estimated using harmonic transition state theory. Transitions are selected and the clock advanced according to the kinetic Monte Carlo algorithm. Unlike traditional applications of kinetic Monte Carlo, the atoms are not assumed to sit on lattice sites and a list of all possible transitions need not be specified beforehand. Rather, the relevant transitions are found on the fly during the simulation. A multiple time scale simulation of Al(100) crystal growth is presented where the deposition event, occurring on the time scale of picoseconds, is simulated by ordinary classical dynamics, but the time i...

Lithium Insertion in Nanostructured TiO2(B) Architectures

Electric vehicles and grid storage devices have potentialto become feasible alternatives to current technology, but only if scientists can develop energy storage materials that offer high capacity and high rate capabilities. Chemists have studied anatase, rutile, brookite and TiO2(B) (bronze) in both bulk and nanostructured forms as potential Li-ion battery anodes. In most cases, the specific capacity and rate of lithiation and delithiation increases as the materials are nanostructured. Scientists have explained these enhancements in terms of higher surface areas, shorter Li(+) diffusion paths and different surface energies for nanostructured materials allowing for more facile lithiation and delithiation. Of the most studied polymorphs, nanostructured TiO2(B) has the highest capacity with promising high rate capabilities. TiO2(B) is able to accommodate 1 Li(+) per Ti, giving a capacity of 335 mAh/g for nanotubular and nanoparticulate TiO2(B). The TiO2(B) polymorph, discovered in 1980 by Marchand and co-workers, has been the focus of many recent studies regarding high power and high capacity anode materials with potential applications for electric vehicles and grid storage. This is due to the materials stability over multiple cycles, safer lithiation potential relative to graphite, reasonable capacity, high rate capability, nontoxicity, and low cost (Bruce, P. G.; Scrosati, B.; Tarascon, J.-M. Nanomaterials for Rechargeable Lithium Batteries. Angew. Chem., Int. Ed.2008, 47, 2930-2946). One of the most interesting properties of TiO2(B) is that both bulk and nanostructured forms lithiate and delithiate through a surface redox or pseudocapacitive charging mechanism, giving rise to stable high rate charge/discharge capabilities in the case of nanostructured TiO2(B). When other polymorphs of TiO2 are nanostructured, they still mainly intercalate lithium through a bulk diffusion-controlled mechanism. TiO2(B) has a unique open crystal structure and low energy Li(+) pathways from surface to subsurface sites, which many chemists believe to contribute to the pseudocapacitive charging. Several disadvantages exist as well. TiO2(B), and titania in general, suffers from poor electronic and ionic conductivity. Nanostructured TiO2(B) also exhibits significant irreversible capacity loss (ICL) upon first discharge (lithiation). Nanostructuring TiO2(B) can help alleviate problems with poor ionic conductivity by shortening lithium diffusion pathways. Unfortunately, this also increases the likelihood of severe first discharge ICL due to reactive Ti-OH and Ti-O surface sites that can cause unwanted electrolyte degradation and irreversible trapping of Li(+). Nanostructuring also results in lowered volumetric energy density, which could be a considerable problem for mobile applications. We will also discuss these problems and proposed solutions. Scientists have synthesized TiO2(B) in a variety of nanostructures including nanowires, nanotubes, nanoparticles, mesoporous-ordered nanostructures, and nanosheets. Many of these structures exhibit enhanced Li(+) diffusion kinetics and increased specific capacities compared to bulk material, and thus warrant investigation on how nanostructuring influences lithiation behavior. This Account will focus on these influences from both experimental and theoretical perspectives. We will discuss the surface charging mechanism that gives rise to the increased lithiation and delithiation kinetics for TiO2(B), along with the influence of dimensional confinement of the nanoarchitectures, and how nanostructuring can change the lithiation mechanism considerably.